PHYSICAL REVIEW B 88, 024114 (2013)
Piezoelectric properties of twinned ferroelectric perovskites with head-to-head
and tail-to-tail domain walls
P. Ondrejkovic,1 P. Marton,1 M. Guennou,1 N. Setter,2 and J. Hlinka1
1
Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 18221 Praha 8, Czech Republic
2
Ceramics Laboratory, Swiss Federal Institute of Technology, Lausanne CH-1015, Switzerland
(Received 13 May 2013; published 22 July 2013)
Longitudinal piezoelectric coefficient of a twinned ferroelectric perovskite material with an array of partially
compensated head-to-head and tail-to-tail 90-degree domain walls has been studied by phase-field simulations in
the framework of the Ginzburg-Landau-Devonshire model of BaTiO3 ferroelectric. In particular, it is shown that
the magnitude of the build-in extrinsic charge at the domain wall and the nanoscale domain size can both promote
rotation of the static polarization vector within the body of adjacent domains. This polarization rotation drives
the domain closer to an orthorhombic state, and the proximity to this ferroelectric-ferroelectric phase transition
is directly responsible for the enhancement of the properties. Our simulations and the theory also suggest that
the same system with nominally overcompensated charged walls may show a negative effective longitudinal
piezoelectric coefficient. The obtained results can be used for quantitative estimates of piezoelectric properties
of domain-engineered crystals.
DOI: 10.1103/PhysRevB.88.024114
PACS number(s): 77.80.Dj, 77.84.−s
I. INTRODUCTION
A wide range of practical instruments exploits the excellent
piezoelectric properties of perovskite ferroelectrics. It is
believed that the organization and the density of non-180degree domain walls in these materials has a sizable impact
on their electromechanical properties.1–4 In principle, this
gives a considerable opportunity for tailoring piezoelectric
properties of a given ferroelectric material by modification of
its domain structure.5 With this perspective, the macroscopic
piezoelectric response of the basic plausible domain configurations has been already analyzed by various theoretical
approaches. In particular, the laminate domain structures
in the tetragonal ferroelectric phase have been analyzed in
the framework of effective-medium theories based on the
analysis of the electromechanical matching of homogeneous
domains,6,7 but also in the framework of phase-field models
allowing to treat size-dependent effects.4,8,9 Several recent
works have drawn attention to the extraordinary possibilities
of incorporating the so-called head-to-head (HH) and tail-totail (TT) domain walls.9–13 These domain walls have been
observed in real materials but they are nominally charged
so that their existence requires some charge compensation
mechanism. Since ferroelectric perovskites can be usually
considered as wide-band semiconductors, it seems likely
that under convenient conditions, the compensation could be
provided directly through the charge carriers from valence
or conduction bands, for example, due to the band-bending
mechanism discussed recently in Refs. 10–12.
Another possibility, advocated in Ref. 13, suggests that the
compensation could be provided by fixed extrinsic charges
related to some ionic defects, such as oxygen or even cationic
vacancies in ABO3 perovskite crystal lattice, segregated at the
domain wall. Interestingly, results of Ref. 13 suggest that the
piezoelectric properties could be considerably enhanced when
the charge compensation by ionic defects is actually quite far
from the ideal nominally required amount. The fact that HH
and TT domain walls can survive in the lattice even when
1098-0121/2013/88(2)/024114(9)
the compensating charge densities deviate from nominally
required values is exploited also in this work. We consider that
the degree of compensation can be modified in experiments by
varying slightly the chemical composition and concentrations
of available defects per domain wall area or by modifying
the condition of the annealing process during which the ionic
defects are likely to diffuse towards the charged domain walls.
But we do not discuss this process here. Our aim is to provide
direct insight to the enhancement of piezoelectric properties of
the twinned BaTiO3 , predicted in Ref. 13 by complementary
analytical and phase-field simulations using the GinzburgLandau-Devonshire (GLD) model with the parameters of
Refs. 4 and 14.
II. TECHNICAL DETAILS
Phase-field simulations presented in this article are based
on the phenomenological GLD model4,15 in which the excess
Gibbs free-energy functional F is expressed in terms of
lowest-order polynomials of ferroelectric polarization Pi ,
its spatial derivatives Pi,j = ∂Pi /∂xj and strain components
eij = (∂ui /∂xj + ∂uj /∂xi )/2, i,j = 1 − 3 as
(1)
F = drfGLD [{Pi ,Pi,j ,eij }] + Fdip [{Pi }],
where the first term
024114-1
fGLD [{Pi ,Pi,j ,eij }]
(e)
(e)
Pi2 + α11
Pi4 + α12
Pi2 Pj2
= α1
i
+ α111
i
i>j
Pi4 Pj2 + Pj4 Pi2
Pi6 + α112
i
i>j
+ α123 P12 P22 P32
− Pi Ei
1
1
+ Cij kl eij ekl − qij kl eij Pk Pl + Gij kl Pi,j Pk,l
2
2
(2)
©2013 American Physical Society
ONDREJKOVIC, MARTON, GUENNOU, SETTER, AND HLINKA
stands for the usual GLD functional. The greek letter symbols
are parameters defining the fully clamped Landau-Ginzburg
potential at a given temperature, Cij kl ,qij kl , and Gij kl stand for
components of elastic, electrostriction, and gradient tensors,
respectively, and Ei stands for the component of applied
electric field, which acts as an independent thermodynamical
variable of the local Gibbs function. The other term in Eq. (1)
describes the electrostatic energy16,17
1
Fdip [{Pi }] = −
(3)
dr[Edip (r) · P(r)],
2
associated with the long-range interaction of individual dipoles
with the electric fields of all other dipoles, expressed via
the inhomogeneous depolarization field Edip created by the
inhomogeneous part of the field of polarization:
P(r ) 3[P(r ) · R]R
−1
, (4)
−
dr
Edip (r) =
4π 0 B
|R|3
|R|5
where R = r − r , B is the relative background permittivity of the medium (without the primary order parameter
contribution), and 0 is the permittivity of vacuum. Let us
stress that in the present problem, the field of polarization
is restricted by periodic boundary conditions, which implies
that the long-range spatial average of the electric field Edip
generated by the field of polarization itself is vanishing.
In order to find the equilibrated domain structure under
specified conditions, we have applied the usual phase-field
approach16,18–21 consisting in a simulation of the natural “equilibration” process by numerical solution of the corresponding
time-dependent Ginzburg-Landau equation (TDGL) for the
field of polarization:
∂Pi
δF
,
= −
∂t
δPi
(5)
where is a kinetic coefficient controlling the energy dissipation rate of the system. Mechanical equilibrium is assumed to
be achieved at each instant so that the inhomogeneous strain
field can be eliminated from the energy functional of Eq. (1)
using the corresponding Euler-Lagrange equations.17–19 The
homogeneous average strain was evaluated in a similar way
assuming fixed external stress boundary conditions.
We used phenomenological parameters for roomtemperature BaTiO3 that were selected as follows:4,15 α1 =
(e)
(e)
= 1.701 × 108 Jm5 C−4 , α12
=
− 2.77 × 107 JmC−2 , α11
8
5 −4
9
9 −6
−2.760 × 10 Jm C , α111 = 8.004 × 10 Jm C , α112 =
4.470 × 109 Jm9 C−6 , α123 = 4.910 × 109 Jm9 C−6 , G11 =
51 × 10−11 Jm3 C−2 , G12 = −2 × 10−11 Jm3 C−2 , G44 =
2 × 10−11 Jm3 C−2 , q11 = 14.20 × 109 JmC−2 , q12 =
−0.74×109 JmC−2 , q44 = 3.14×109 JmC−2 , C11 = 27.50 ×
1010 Jm−3 , C12 = 17.90 × 1010 Jm−3 , C44 = 5.43 × 1010
Jm−3 , and B = 7.35. These parameters are the same as those
defined as “model I” (at 298 K) in Ref. 15, except for the
parameter q44 , which has been later corrected by a factor of
two4 . We use a kinetic coefficient = 4 × 104 C2 J−1 m−1 s−1
to define realistic time scales in the simulated processes,22
but this choice has no effect on the results presented here.
Note that the experimental uncertainty of these parameters
is obviously far beyond the numerical precision given here.
Nevertheless, we prefer to provide here the actual values used
in the calculation as this might be of use to those willing to
PHYSICAL REVIEW B 88, 024114 (2013)
reproduce our results. The same holds for most of the derived
quantities given in this paper.
The inherent symmetry of the problem treated in this article
allowed to perform the simulation within a 2D rectangular
discrete array (typically, 128 × 128 individual sites), fulfilling
periodic boundary conditions. Equation (5) was resolved
numerically in Fourier space by a second-order semi-implicit
method with spatial steps 0.5 nm and individual time steps 2 fs.
The piezoelectric properties of a given domain configuration were derived by a straightforward procedure: at first, the
initial structure was relaxed (equilibrated) for about 50–100 ps
at T = 298 K. Then, the relaxation was repeated with that
equilibrated structure under a small applied probing homogeneous electric field. The average of the induced strain was
evaluated for several values of the probing field. Then, the
average strain as a function of the probing field was fitted
to a polynomial function and the linear increment in average
strain was determined. The quadratic increment describing
the electrostriction contribution was negligible for the small
probing fields applied here. Finally, the ratio of the linear
increment and the probing field was used to evaluate the converse (indirect) piezoelectric coefficient. Similar equilibrating
under small homogeneous steplike applied stress was used to
determine the direct piezoelectric coefficient.
III. INVESTIGATED DOMAIN STRUCTURE
In this study, we have investigated the room-temperature
piezoelectric response of domain-engineered BaTiO3 crystal
assuming the particular case of the domain structure sketched
in Fig. 1(a). This domain structure was stabilized by a piecewise homogeneous macroscopic field sketched in Fig. 1(b).
What is the motivation for this choice? Within the GLD
model described above, the lowest energy state of an infinite
sample at ambient stress-free conditions corresponds to the
single-domain tetragonal phase with spontaneous polarization
PS , PS = |PS | = 0.2652 Cm−2 . When such an ideal domain
state is cut by a plane with an outward pointing normal n,
such surface possesses (an uncompensated) surface bound
charge with planar density σS = PS · n. Let us consider a
piece-wise homogeneous polarization configuration shown in
Fig. 1(a) (equivalent to those considered in Ref. 13). It is
composed of alternating (100) and (01̄0) oriented tetragonal ferroelectric domain states of equal thickness separated
by infinitely sharp 90◦ domain walls, perpendicular to the
FIG. 1. (Color online) Schematic representation of initial conditions: (a) simple two-dimensional lamellar domain structure with
charged 90◦ domain walls assumed in present simulations and (b) the
imposed piece-wise homogenous electric field Eσ , associated with
the compensating charge distribution defined by Eq. (7).
024114-2
PIEZOELECTRIC PROPERTIES OF TWINNED . . .
PHYSICAL REVIEW B 88, 024114 (2013)
√
direction y = (1,1,0)/ 2. Because of the alternating HH
and TT domain wall arrangement, these domain boundaries
possess planar charge densities σdep = 2σS = ±2|PS · y | (HH
domain wall is positively charged). In a material with a
background permittivity B , this charge distribution would
create an inhomogeneous depolarization field
Edep
(PS · y )y
=−
.
B 0
(6)
It is well known that in the absence of additional chargecompensation mechanisms, such a huge depolarization field
(about 2.7 GV/m in the room-temperature BaTiO3 ) would
destroy the assumed domain structure, and indeed, it is what
happens when the structure of Fig. 1(a) is used as an initial
configuration in phase-field simulations.
As already mentioned, plausible carrier species involved
in the charge compensation of the bound charge density σdep
are either the highly mobile electrons and holes available at
domain walls due to electronic band-bending mechanism,10–12
or much less mobile ionic defects segregated at these domain
walls, such as the oxygen or cationic vacancies or other
charged defects.13 While the electronic subsystem could in
principle follow the piezoelectric response, the compensation
ionic charge density can be regarded as a fixed static charge
distribution. For simplicity, we shall consider the latter case
here, even though with some caution, the present results can be
also used for gaining some additional insight into the former
case as well (e.g., in combination with the analysis of Ref. 12).
It has been shown13 that the HH and TT walls may survive
even when the magnitude of the imposed compensation charge
density considerably differs from σdep . Therefore we shall
express the imposed compensation density as
σ = −ρ ∗ σS = − 12 ρ ∗ σdep .
(7)
Here we have used, along with the Ref. 13, a dimensionless
indicating the imposed compencharge density parameter ρ ∗ , √
sation density in units of PS / 2. For simplicity, the imposed
compensation charge density is assumed to be of the same
magnitude for both HH and TT walls. The inhomogeneous
electric field Eσ caused by the imposed compensation charge
density σ is directed opposite to the nominal depolarization
field Edep , and it can be expressed as Eσ = − 12 ρ ∗ Edep [see
Fig. 1(b)]. Therefore the exact compensation of the nominal
bound charge density σdep and of the nominal Edep takes place
for ρ ∗ = 2. Therefore the electric field E of Eq. (2) applied in
the present phase-field simulations consists of (i) zero or small
homogenous probing field and (ii) a rather strong, piece-wise
homogeneous electric field Edep , describing the effect of the
imposed charge carriers.
Obviously, in our phase-field simulations, polarization is
allowed to evolve under TDGL equation (5), so that the domain
walls have finite thicknesses. Moreover, the polarization within
the body of ferroelectric domains differs from spontaneous
single domain value of PS due to the influence of the following
three aspects: (i) the very presence of domain walls, (ii) the
domain wall distance (domain size effect), and (iii) the imposed fixed compensation charge density σ . Consequently, the
actual depolarization field due to the equilibrium polarization
distribution can be considerably different from the nominal
value of Eq. (6). As we shall see, the field of polarization
can, in fact, adjust to a broad range of values of ρ ∗ without
destroying the basic pattern of HH and TT domain wall array.
IV. RESULTS
The model for BaTiO3 that we used in our previous
works4,14 is somewhat different from the one used in Ref. 13,
and so it is interesting to compare the predictions of both
models for an equivalent domain structure. For this purpose,
we have chosen the set of data obtained for the domain arrangement shown in Fig. 1(a) with the 22.6-nm-thick domains.
The comparison of the data obtained for the longitudinal
[100]
piezoelectric coefficient d33
is shown in Fig. 2. In both
cases, data show a dramatic increase in the response when
ρ ∗ is significantly smaller than 2. Moreover, we have extended
the simulations also to the case when the ρ ∗ parameter is
significantly larger than 2. In this case, a marked dependence
is also observed, but very interestingly, the material then shows
[100]
a negative longitudinal piezoelectric coefficient d33
.
All data indicated in Fig. 2 were obtained as a converse
piezoelectric response, i.e., from linear expansion of the material in the [100] direction in response to a small homogeneous
probing bias electric field (100 V/m) in the same direction.
We have checked that the effect was linear in the whole
investigated range of parameters ρ ∗ (explicit electrostriction
contribution was negligible). In addition, we also simulated
the direct piezoelectric effect (polarization induced by a small
uniaxial stress applied in the [100] direction) and obtained
same values.
One of the obvious advantages of the phase-field simulations is that the calculation can automatically take into account
the finite size effects related to the polarization correlation
lengths (natural domain wall thickness). Therefore we have
also investigated the influence of the domain size on the
[100]
FIG. 2. Effective longitudinal piezoelectric coefficient d33
of
the BaTiO3 ferroelectric material with a hypothetic domain structure sketched in Fig. 1(a) as a function of the compensation
charge parameter ρ ∗ (for 22.6-nm-thick domains). Values obtained
from the present phase-field calculations (open circles) can be compared to the values of the phase-field calculations given in Table III
of Ref. 13 (full circles). In both cases, the values were determined
from the overall elongation of the material in the [100] direction
in response to a small probing electric field in the same direction
(simulation of a converse piezoelectric response).
024114-3
ONDREJKOVIC, MARTON, GUENNOU, SETTER, AND HLINKA
PHYSICAL REVIEW B 88, 024114 (2013)
[100]
FIG. 3. Effective longitudinal piezoelectric coefficients d33
as
a function of domain wall distance wD for selected values of
compensation charge parameter ρ ∗ . Open point symbols refer to this
study, full circles are data from Table IV of Ref. 13. Full lines are
guides for eye, the dotted line indicates the analytically calculated
[100]
piezoelectric coefficient for ρ ∗ = 2 case
limiting value of the d33
(see in Sec. V A).
macroscopic piezoelectric response of our system. Similarly,
as in Ref. 13, we have found that decreasing domain size can
markedly enhance the piezoelectric properties (see Fig. 3).
At the same time, we have found a much smaller effect
in the reported case of ρ ∗ = 1.9 (see full and open point
symbols in Fig. 3). We have noticed that a comparable or
even greater enhancement can be still achieved by decreasing
ρ ∗ rather than the domain wall distance, what could actually
turn out to be more feasible in experiments (compare the open
circle and open triangle symbols in Fig. 3). Surprisingly, we
have observed that the longitudinal piezoelectric coefficient
[100]
d33
can also decrease with decreasing domain size (e.g., for
ρ ∗ = 1.6, see open triangle symbols in Fig. 3).
In order to get more insight into the mechanism of these
effects, it is interesting to explore the local contributions to the
calculated piezoelectric response. For this purpose, we have
analyzed the local change of the field of polarization induced
by a small homogeneous probing uniaxial stress along the
[100] direction (σxx = 104 Pa, other stress tensor components
were zero). Local polarization changes along [100] and [010]
directions (δPx and δPy ) in the vicinity of the HH domain
wall are shown in Figs. 4(a) and 4(b), respectively. Obviously,
[100]
the longitudinal piezoelectric coefficient d33
= dxxx can
be evaluated as the spatial average of δPx over the whole
specimen, divided by the applied stress σxx . One can thus easily
see from Fig. 4(a) that the huge sensitivity of the longitudinal
piezoelectric coefficient is related to changes in the B domain,
i.e., in the domain having its nominal spontaneous polarization
perpendicular to the applied stress.
Let us stress that the calculated domain walls (i.e., kinks in
the spatial distribution of the ferroelectric polarization) are not
a priori fixed in our simulations but they are strongly attracted
to the rigidly fixed build-in charge layers. Therefore domain
wall displacements under the probing fields are negligibly
small. It could be also understood from Fig. 4, which shows
that the contribution to the piezoelectric response is not
FIG. 4. (Color online) Local polarization increment induced by a
small, x-axis tensile uniaxial stress (σxx = 104 Pa), traced along the
y direction in the vicinity of a HH domain wall. The domain wall
distance is fixed to 22.6 nm, various symbols refer to data calculated
for various values of the charge parameter ρ ∗ , (a) x component of
the polarization increment and (b) y component of the polarization
increment.
significantly enhanced in the domain wall area. In this sense,
domain walls in our simulations may be considered as pinned.
Previously, authors of Ref. 13 proposed that the decrease
of the piezoelectric response with ρ ∗ approaching the value
of 2 (see Fig. 2) might be caused by enhanced domain wall
pinning. However, our results clearly show that pinning is very
strong for all investigated values of ρ ∗ . In reality, our Fig. 4
testifies that this tendency apparent from Fig. 2 is rather caused
by variation of the piezoelectric response within the domain
interior. The reason of this variation will be discussed in the
next section.
V. DISCUSSION
A. Nominally compensated walls (ρ ∗ = 2).
Let us first discuss the ρ ∗ = 2 ideal case when the amount of
the imposed compensation charge exactly cancels the nominal,
bulk depolarization field given by Eq. (6). In the limit of thick
enough domains, the body of the domain is in the normal
bulk state, and it can be described by its usual (equilibrium
single-domain) material tensors. In this case, the effective
macroscopic piezoelectric response of the multidomain state
024114-4
PIEZOELECTRIC PROPERTIES OF TWINNED . . .
PHYSICAL REVIEW B 88, 024114 (2013)
depends only on these material tensors. The resulting effective
values obviously also involve the domain fraction ratio and
the orientation of the domain wall interfaces, as they impose
the classical boundary conditions for otherwise piecewise
homogeneous electrical and mechanical fields. Convenient
formalisms for the calculation of the effective piezoelectric
response of such an ideal laminar domain structure have been
considered previously by several authors (see Refs. 6,7,23
and 24 and earlier works cited there).
The effective piezoelectric tensor for the present domain
structure [shown in Fig. 1(a), with equal domain fractions]
can be most conveniently expressed in the Cartesian system
x ,y ,z, obtained from the natural crystallographic x,y,z
system by 45◦ rotation around the z axis (see Fig. 1).
The explicit formulas for the effective piezoelectric tensor
components dijeff read7
eff
d11
=
d11
−
eff
d12
= d12
−
eff
d13
= d13
−
12
d12
,
22
12
d22
,
22
12
d23
,
22
eff
d35
= d35
,
eff
d26
=
− d12
s13 )
s (d s11
d26
− 36 23
2
s33 s11 − s13
s (−d s + d s )
− 16 23 13 212 33 ,
s33 s11 − s13
(8)
(9)
(10)
(11)
wD = 22.6 nm
wD = 90.5 nm
wD = ∞
eff
eff
d11
= −d22
75.2
68.5
65.9
eff
eff
d12
= −d21
−19.2
−12.7
−10.1
eff
eff
d13
= −d23
−38.8
−38.6
−38.6
eff
eff
d16
= −d26
−25.8
−25.6
−24.7
eff
−d35
−226.4
−224.4
−224.0
eff
d34
=
[100]
is identical to the so far discussed quantity d33
. Therefore the
value from the last column was also included as the expected
asymptotic “thick domain” value in Fig. 3. The data of Fig. 3
also reveal that the domain size has a rather small impact on
[100]
the d33
coefficient in the ρ ∗ = 2 case. In fact, the size effect
observed here is much smaller than the size effect predicted,
e.g., for the uncharged head-to-tail domain-wall structures
(ρ ∗ = 0) studied by us previously in Ref. 4.
B. Nominally uncompensated walls (ρ ∗ = 2)
(12)
where dij , sij , and ij are single-domain material tensor
components in the x ,y ,z reference frame.7 From these expressions, and using the known stress-free room-temperature
single-domain BaTiO3 material tensors corresponding to
our GLD potential expansion in the conventional axis
setting (11 = 3300, 33 = 193, d31 = −40.9 pC/N, d33 =
100 pC/N,d15 = 448 pC/N,s11 = 8.45 × 10−3 GPa−1 ,s12 =
−1.97 × 10−3 GPa−1 , s13 = −5.34 × 10−3 GPa−1 , s33 =
13.33 × 10−3 GPa−1 ,s44 = 25.28 × 10−3 GPa−1 , and s66 =
18.42 × 10−3 GPa−1 ), we arrived at the following values of the
eff
effective piezoelectric tensor components: d11
= 56.9 pC/N,
eff
eff
eff
d12 = 22.0 pC/N, d13 = −54.6 pC/N, d35 = 316.7 pC/N,
eff
and d26
= 107.6 pC/N. Finally, the effective piezoelectric
tensor was expressed in the original crystallographic frame
x,y,z. The resulting values are displayed in the last column
of Table I. For the sake of comparison, the full macroscopic
piezoelectric tensor was also determined from phase-field
simulations for wD = 22.6 nm and wD = 90.5 nm (ρ ∗ = 2,
see Table I). The agreement between the effective medium
theory and the wD = 90.5 nm phase-field calculations is nice
and suggests that, in this case, the 100-nm-thick domains could
behave fairly much the same as the macroscopic domains.
Obviously, the piezoelectric component shown in the first
row of Table I,
1 eff
eff
eff
eff
,
= √ d11
+ d12
+ d26
d11
2 2
TABLE I. Cartesian components of the macroscopic piezoelectric
tensor (in the Voigt notation with respect to the natural pseudocubic
system x,y,z and in pC/N units) of the multidomain structure shown
in Fig. 1 with an ideal nominal compensation (ρ ∗ = 2). First and
second numerical columns refer to data from phase-field simulations,
the last column gives the prediction of the effective-medium theory
relevant to the limit of thick domains.
(13)
The present phase-field simulations imply that the most
[100]
spectacular enhancement of the d33
coefficient is achieved
by a variation of the parameter ρ ∗ (ρ ∗ = 2). Since the size
effect seems to be a secondary effect, it can be expected
that the piezoelectric enhancement is related to the modified
response of the body of the domains, rather than to an unusual
response of domain walls. Detailed inspection suggests that the
variation of ρ ∗ indeed induces a notable change of the equilibrium polarization vector in the body of the domain. This change
can be roughly described as a polarization rotation around
the z axis. A similar rotation was present in previous studies
of piezoelectric response of domain-engineered BaTiO3 .4,8,12
In fact, the general tendency to the polarization rotation in
BaTiO3 and similar materials is known to be related to the large
value of “transverse” permittivity element 11 (i.e., 11 /33 1).2,12,25 What needs to be elucidated in this particular case
is (i) the driving force inducing this static rotation and
(ii) whether this rotation is directly involved in the enhancement of the considered piezoelectric coefficient.
In our case, the polarization vector of A and B domains
tends to rotate in opposite sense (see Fig. 5). To quantify
this effect, we have introduced angles αA and αB , describing
the deviation of the equilibrium polarization in the center
of domains A and B from their nominal orientation [shown
in Fig. 1(a)]. The αA and αB angles were read out from
the simulated patterns and it was confirmed that they are
equal for all the above considered cases [the angles were
considered positive for the situation indicated in Fig. 5(b)], so
that the domain structure maintains the original C2v (mz my 2x )
macroscopic symmetry. The equilibrium polarization is found
to be tilted towards the domain wall interface for ρ ∗ < 2
and away from it for ρ ∗ > 2, respectively (see Fig. 5). This
024114-5
ONDREJKOVIC, MARTON, GUENNOU, SETTER, AND HLINKA
FIG. 5. (Color online) Schema indicating the observed tilt of the
equilibrium polarization P(A) and P(B) in the middle of the domain
A and B, respectively. The equilibrium polarization is tilted towards
the domain wall interface for ρ ∗ < 2 as shown in (a) and away from
it for ρ ∗ > 2, as shown in (b). The role of the αA and αB angles is
discussed in the main text.
tendency could be very easily understood: a positive angle
implies enhancement of the normal component of polarization,
and thus it also increases the depolarization field created by
the mismatch of the normal components of the polarization in
domains A and B. This is precisely what has to be arranged to
match the excess of the charge density (ρ ∗ > 2). Similarly,
negative angles lead to smaller mismatch of the normal
components of the polarization in domains A and B, what
is needed to accommodate the situations with small imposed
compensation charge densities (ρ ∗ < 2).
[100]
The correlation of the d33
piezoelectric coefficient with
the tilt angle αA = αB is apparent from Fig. 6. The values are
somewhat spread by the dependence on the domain size, but it
is clear that the overall dependence is strongly nonlinear and
[100]
the most spectacular enhancement of the d33
is observed
[100]
for the αA ≈ −20◦ , and the strongly negative d33
values are
obtained for αA ≈ +20◦ . These critical polarization tilt angles
correspond to the polarization pointing roughly in the middle
between [100] and [110] crystallographic directions. It clearly
demonstrates that the observed enhancement is related to
the competition between tetragonal and orthorhombic ground
states, inherent to the room-temperature GLD potential of
BaTiO3 .4,12,26,27
PHYSICAL REVIEW B 88, 024114 (2013)
FIG. 7. Phase diagram showing stability domains of distinct types
of domain structures obtained in present phase-field simulations for
a range of parameters wD and ρ ∗ , defining the domain wall distance
and the imposed compensating charge density, respectively. Point
symbols are determined from the divergence of the piezoelectric
response as a function of ρ ∗ (for selected fixed values of wD ), lines
are just guides for eye. Regions II and III can be considered as the
original domain structure of Fig. 1(a), only with slight symmetric
tilts of the polarization, as shown in Figs. 5(a) and 5(b), respectively.
Shaded areas I and IV correspond to a lower-symmetry domain
structure (monoclinic one), with unequal polarization tilts, sketched
in Figs. 8(a) and 8(b). In the region V, the polarization is perpendicular
to the imposed charge planes and the domain walls are transformed to
180-domain ones with HH and TT arrangements. See text for detailed
explanation.
To shed additional light on this phenomenon, we have
explored an even broader range of parameter ρ ∗ and realized
that for a certain range of parameters, the structure with
αA = αB is no more stable. The resulting stability map as
a function of domain density 1/wD and the extrinsic charge
density parameter ρ ∗ is sketched in Fig. 7. Indeed, only
regions II and III correspond to the symmetric domain structure
(αA = αB ) with domain states depicted in Figs. 5(a) and 5(b),
respectively. In contrast, an asymmetric domain structure (with
αA = αB ) has lower energy than the symmetric one when ρ ∗
and wD parameters are kept within areas denoted as I and IV.
Indeed, in region I, either the A- or B-domain state polarization
is almost parallel to the domain wall, while in region IV, either
the A or B-domain state polarization is almost perpendicular
to the domain wall (see Fig. 8). In the region V, the structure
(degrees)
FIG. 6. (Color online) Effective longitudinal piezoelectric coef[100]
(wD ,ρ ∗ ) as a function of the deviation angle, joining the
ficients d33
points with equal domain thicknesses. Dashed line was calculated
from the Eqs. (15) and (22), derived in Sec. V C.
FIG. 8. (Color online) Schematic image indicating the observed
tilt of the equilibrium polarization P(A) and P(B) in the middle of
the domains A and B, respectively, as obtained in our phase-field
simulations with input parameters falling (a) in the region I and (b)
in the region IV of Fig. 7.
024114-6
PIEZOELECTRIC PROPERTIES OF TWINNED . . .
PHYSICAL REVIEW B 88, 024114 (2013)
is again symmetric, but the polarization in both domains is
perpendicular to the domain wall (αA = αB = π/4). It is easy
to verify, that these geometrical arrangements provide the
suitable step of the normal component of the polarization
vector at the domain wall, needed for near-compensation for
the imposed charge density ρ ∗ . In fact,
√ we have seen that
[P(A) − P(B)] · y differs from ρ ∗ PS / 2 by less than 1%.
It is obvious from Fig. 7. that the large magnitude of the
piezoelectric response shown in Figs. 2 and 6 is associated
with the borderline between regions I and II and the borderline
between regions III and IV, respectively. The enhancement
might be discussed even from the point of view of the Landau
theory of symmetry breaking phase transition since at these
borderlines, the domain structure is lowering its symmetry. It is
worth noting that there is no truly single-domain orthorhombic
state in our “phase diagram,” since the imposed electric field
due to the ρ ∗ charge density induces a periodic wave that
breaks the translation symmetry of the problem. In that sense,
the structure is actually quite reminiscent of incommensurate
dielectrics, where the enhancement of permittivity is also
frequently found near various almost continuous transitions
between modulated ferroelectric phases.28,29
In Fig. 6, data points calculated for equal domain wall sizes
are connected with a common continuous line and plotted
with the same symbols. Therefore one can see that there
are systematic deviations related to the size effects. To some
extent, the slope of the borderline between regions I and II in
Fig. 7 allows to understand the spread of the data shown in
Fig. 6. As could be guessed, the domain size effect is mainly
caused by a partial charge compensation within the domain
wall, which, obviously, plays a role only when domain sizes
are of the order of the domain wall thickness.
C. Mechanism of the piezoelectric enhancement
Finally, it remains to understand how the polarization
[100]
rotation influences the d33
coefficient. For this purpose, we
shall discuss only the regions II and III here. One can see from
[100]
Fig. 4(a) that the strongest enhancement of the d33
coefficient
is originating from the response of the domain B. Therefore
let us investigate the properties of a homogenous domain
state B under the bias depolarization field. For simplicity, we
shall assume that this depolarization field results in a strict
rotation of the spontaneous polarization by the angle α = αB
(its influence on the magnitude of the spontaneous polarization
vector is neglected). Then, the spontaneous polarization can
be written as P(B) = (−P sin α, − P cos α,0). Under these
assumptions, the permittivity tensor of domain B can be
obtained from the Hessian of the stress-free Landau energy f ,
−1
2
∂ f
ij =
.
(14)
∂Pk ∂Pl
For a qualitative understanding, it is sufficient to keep only
the lowest-order terms in α. Recently, it has been explained
that the static rotation of the spontaneous polarization greatly
enhances the transverse susceptibility component xx (α).12 A
similar result is recovered here:
11
,
(15)
xx (α) ≈
1 − Kα 2
with
K = −11 L + 411 33 M 2 ,
L = 12a11 PS2 − 2a12 PS2 + 8a112 PS4 ,
M=
2a12 PS2
+
(16)
4a112 PS4 .
This formula allows to explain12 that for K > 0 the
permittivity xx actually diverges as the angle approaches the
critical value, where the denominator of the formula goes
to zero. Using the adopted Landau-Devonshire expansion
parameters given above and the stress-free Landau coefficients
a11 , a12 defined as15,26
2
2 1 q̂11
q̂22
(e)
,
α11 = α11 −
+2
6 Ĉ11
Ĉ22
(17)
2
2
2 q̂11
1
q̂22
q44
(e)
,
2
α12 = α12 −
−2
+3
6
C44
Ĉ11
Ĉ22
where
Ĉ11 = C11 + 2C12 ,
Ĉ12 = C11 − C12 ,
q̂11 = q11 + 2q12 , q̂12 = q11 − q12 ,
(18)
allows to calculate the parameters in the above equations
.
.
as K = 15.7, L = −4.15 × 108 m F−1 , and M = 1.34 ×
8
−1
10 m F . In this case, the critical angle at which the
2
denominator vanishes, Kαcrit
≈ 1 is close to 15◦ , in a rough
agreement with the divergence shown in Fig. 6. Obviously, the
small value of this angle is directly related to the presence of
subsidiary potential minima in Landau potential, associated
with the metastable orthorhombic phase.12
This depolarization-field-induced divergence of the
transversal susceptibility is responsible also for the enhance[100]
ment of the d33
piezoelectric coefficient. In the absence
of external stresses, the equilibrium homogenous strain is
determined by electrostriction coupling
exx = Q11 Px2 + Q12 Py2 + Pz2 ,
(19)
where Q11 = 0.11045 m4 C−2 , Q12 = −0.0452 m4 C−2 (see
Ref. 15 and works cited therein). The longitudinal piezoelectric
coupling along the direction x (in domain state B) can be
therefore expressed as
dxxx (α) =
∂exx
∂Ex
(20)
leading to
dxxx = −2Q11 PS xx sin α − 2Q12 PS xy cos α.
(21)
Note that the first term indeed involves the divergent
susceptibility component xx . The second term contains only
the off-diagonal permittivity component xy , which for small
values of the angle α reads xy ≈ −211 33 Mα, and, therefore,
this second term does not promise any anomalous property
enhancement. Numerical calculation of the dxxx (α) according
to the Eq. (21) shows that the first term indeed reproduces
the critical trend obtained from phase-field simulations. In
fact, since domain B represents just one half of the sample
024114-7
ONDREJKOVIC, MARTON, GUENNOU, SETTER, AND HLINKA
[100]
volume, the macroscopic piezoelectric coefficient d33
for
large domains can be fairly well estimated as
[100]
eff
≈ d11
+ 0.5dxxx ,
d33
(22)
eff
where d11
is taken from Eq. (13) and dxxx is taken from
Eq. (21). The resulting theoretical dependence is shown by
the dashed line in Fig. 6. The nice agreement of this simple
theory with the phase-field simulations for large domain sizes
demonstrates clearly that the bulk polarization-rotation mechanism described by Eqs. (15) and (21) is really responsible
for the strongest piezoelectric enhancements simulated in this
work as well as in the work of Ref. 13.
VI. CONCLUSION
In summary, in this study we have used a phase-field
approach to simulate piezoelectric response of domainengineered BaTiO3 single crystal with a regular array of
mechanically compatible 90◦ charged domain walls. Charged
domain walls were stabilized by fixed planar interfacial
charge located at their equilibrium positions. Assumed domain
configuration consists of alternating HH and TT-type parallel
boundaries. It was seen that both the domain period and
the magnitude of the stabilizing interfacial charge (expressed
through the dimensionless parameter ρ ∗ ) influences the macroscopic piezoelectric response of such domain engineered
material. It was found that the parameters ρ ∗ and wD also
control quite sizable changes in the equilibrium field of
polarization in the whole volume of the sample; mapping of
the stable polarization configurations revealed stability regions
of several qualitatively different states in the investigated
range of these parameters (shown in Fig. 7). In fact, the
strongest modification of the piezoelectric response occurs
near boundaries of these stability regions.
Remarkably, we have seen that the longitudinal piezoelec[100]
tric coefficient d33
associated with the pseudocubic direction
[100] can attain magnitudes of the order of 1000 pC/N.
Phase-field simulation revealed that this tremendous increase
originates from the piezoelectric response of the normally
piezoelectrically inactive domain state B (with nominal
polarization perpendicular to the probed direction [100]).
Complementary analytical calculations demonstrate that this
enhancement is a consequence of the rotation of the spontaneous polarization within the domain B and the presence
of free-energy minima associated with the energetically
comparable metastable orthorhombic state. In addition, these
calculations result in explicit formulas allowing to estimate this
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It is likely that in a material with build-in charged layers
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In summary, we have explored here the piezoelectric properties of ferroelectric perovskites with nominally charged twin
boundaries. The previous efforts to realize domain-engineered
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mostly aimed to decrease the domain size.3 On the other
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more straightforward way to approach the critical polarization
rotation angle. Moreover, this mechanism holds even for
large domains. Therefore this work hints towards a very
different possibility for domain engineering. It seems that here
described prospects of the property enhancement by build-in
charged layers are extremely challenging.
ACKNOWLEDGMENTS
The work was supported by the Sciex-NMSch program
of the Swiss Federal Government (Project Code 11.116).
Authors are grateful to A. K. Tagantsev for useful suggestions
and discussions and very detailed and critical reading of
the manuscript. The research leading to these results has
received funding from the European Research Council under
the EU 7th Framework Programme (FP7/2007-2013)/ERC
Grant Agreement No. 268058.
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